# Computation Workshop Solution Checker

## The Fibonacci Sequence

The sequence $$1, 1, 2, 3, 5, \ldots$$ is called the Fibonacci Sequence. It is the sequence $$f(1)$$, $$f(2)$$, $$f(3)$$, $$f(4)$$, $$\ldots$$ defined by

• $$f(1) = 1$$,
• $$f(2) = 1$$, and
• $$f(n + 2) = f(n + 1) + f(n)$$ for all $$n \ge 1$$.

So for example $$f(12) = 144$$. So the first two digits of $$f(12)$$ are $$14$$ and the last two digits of $$f(12)$$ are $$44$$.

Part A What are the first three digits of $$f(123)$$? What are the first six digits of $$f(123456)$$? What are the last six digits of $$f(123456)$$? What are the first nine digits of $$f(123456789)$$? What are the last nine digits of $$f(123456789)$$?

## Digital Self Powers

The self-power of a number $$x$$ is defined to be $$x^x$$. The digital-self-power of a number is defined to be the sum of the self powers of all of its digits (in base 10). For example, the digital-self-power of $$402$$ is $$4^4 + 0^0 + 2^2 = 256 + 1 + 4 = 261$$.

Part A Find an integer which is one more than its digital-self-power. Find an integer (greater than 1) which equals its digital-self-power Find a positive integer which is one less than its digital-self-power Find the largest integer which is at most one away from its digital-self-power